How does this compute exact dice probabilities?

It builds the distribution one die at a time by convolution: it starts with the modifier, then for each die it spreads the current distribution across that die's faces and sums overlapping totals. This gives exact rational probabilities, not a simulation.

What notation can I use?

Standard RPG notation: NdM for N dice of M sides, joined with plus or minus, plus an optional flat modifier. Examples are 3d6, 2d8+1d4, 4d6-2, and 1d20+5. Spaces are ignored and dice may be listed in any order.

Why is the middle total most likely?

Summing several dice concentrates results toward the average because there are many more combinations that produce a middle total than an extreme one. With 3d6 there is only one way to roll 3 but 27 ways to roll a 10 or 11, so the distribution peaks in the centre.

What do at-least and at-most odds mean?

At-least is the cumulative probability of rolling that total or higher, which answers questions like the chance to beat a target number. At-most is the probability of that total or lower. Both are shown for every total so you can read success odds directly.

Is there a limit on pool size?

The tool caps the number of dice and total face range to keep the table fast and readable in the browser. Very large pools (hundreds of dice) are rejected, but any realistic RPG pool computes instantly because convolution is efficient.

Custom Dice Pool Probability Table

This tool gives the exact probability of every possible total for any dice pool, using combinatorial math rather than simulation. Enter notation like 3d6+2 or 2d8+1d4 and read off the full distribution, the average, and cumulative success odds.

How it works

The distribution is built by convolution, one die at a time. Each die spreads the running distribution across its faces and re-sums the overlaps:

start: probability 1.0 at the flat modifier
for each die dM:
    new[t + face] += old[t] / M   for face = 1..M

Because every face is equally likely and independent, this produces the exact probability of each total. The chance of any total is the count of dice combinations that sum to it divided by the total number of combinations.

Example and notes

For 3d6 there are 216 equally likely outcomes. Only one of them sums to 3 (all ones), so P(3) is 1/216 ≈ 0.46 percent, while 27 combinations sum to 10 or 11, making each about 12.5 percent — the peak of the classic bell-shaped 3d6 curve. The at-least column then tells you, for instance, the chance to roll 13 or more. Use the mean to balance damage or difficulty and the cumulative columns to set target numbers that hit a desired success rate.